This is a PICS project (Projet International de Coopération Scientifique) from CNRS. This project consists of a concerted attack to a number of fundamental dynamical systems sharing a significant core of characteristics, not least a type of algebraic nature in that they can be represented by some rational vector field on an algebraic manifold. Among the systems and the problems targeted by us, there are: the geodesic completeness of Lorentz manifolds; Painlevé equations (dynamics, special solutions, and integrability); Dynamics of higher dimensional Halphen systems and connected topics including the SU(2)-symmetric ASD equation (relation with P6, their (in)completeness, their Twistor spaces) and the Gauss-Manin connection in the Dwork family. Albeit it is not clear at first glance, all these topics are deeply interconnected many new ideas can be brought to the area by building on a few – a priori unrelated - papers.